In the research project, which is to be composed of two substantively and logically connected papers, the authors form such a conceptual framework, that enables characteristic of fractal structures from the point of view of philosophical concept of emergence. In the first part, they present main ideas of the philosophy of emergence as well as they attempt at capturing emergent units in the process of fractals' generating. However, they maintain classical understanding of the relation in question. In the second part, due to demonstration of weaknesses of classical accounts as insufficient in specific context of mathematical structures under scrutiny, the authors show that the discourse about emergence in mathematics becomes meaningful and valid trough adaptation of quasi-empiric approach towards some issues in mathematics, approach grounded in philosophy of formal sciences.